In this subsection we have constructed the geometric Seiberg–Witten map for noncommutative gauge
theories with gauge group \(U(N)\)  or \(GL(N)\)  (or products thereof) and
with \(\star \) -product given by a general abelian
twist (REF ). For abelian gauge groups the
Seiberg–Witten map can be constructed for any
\(\star \) -product associated with an arbitrary Poisson tensor.
The map is obtained in [1]} using Kontsevich formality theorem [2]}.
The study of its global geometric aspects shows that the
Seiberg–Witten map quantizes line bundles with connections on a
Poisson manifold to quantum (noncommutative) line bundles with noncommutative
connections [3]}
The Seiberg–Witten map for nonabelian gauge groups and with arbitray
Poisson tensors is in general an open problem, we refer to [4]}
for interesting insights.
The global geometric aspects of the Seiberg–Witten map are well understood for nonabelian \(U(n)\) -gauge
fields on noncommutative tori: the Seiberg–Witten map defined in
(), (), with \(\gamma =-3\) , \(\rho =i\) , quantizes vector bundles on tori
with connections to vector bundles on noncommutative tori with
noncommutative connections and the results are nonformal in the sense
that they do not rely on power series expansion in the noncommutativity
parameter \(\theta \) .
